Write as a Single Logarithm 1/3( natural log of 64- natural log of 8x^3)

$\frac{1}{3} (\ln (64) - \ln (8 x^{3}))$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{1}{3} \cdot \ln (\frac{64}{8 x^{3}})$

Step 2

Cancel the common factor of $64$ and $8$ .

Factor $8$ out of $64$ .

$\frac{1}{3} \cdot \ln (\frac{8 \cdot 8}{8 x^{3}})$

Cancel the common factors.

Factor $8$ out of $8 x^{3}$ .

$\frac{1}{3} \cdot \ln (\frac{8 \cdot 8}{8 (x^{3})})$

Cancel the common factor.

$\frac{1}{3} \cdot \ln (\frac{8 \cdot 8}{8 x^{3}})$

Rewrite the expression.

$\frac{1}{3} \cdot \ln (\frac{8}{x^{3}})$

Step 3

Simplify $\frac{1}{3} \ln (\frac{8}{x^{3}})$ by moving $\frac{1}{3}$ inside the logarithm.

$\ln ({(\frac{8}{x^{3}})}^{\frac{1}{3}})$

Step 4

Apply the product rule to $\frac{8}{x^{3}}$ .

$\ln (\frac{8^{\frac{1}{3}}}{{(x^{3})}^{\frac{1}{3}}})$

Step 5

Simplify the numerator.

Rewrite $8$ as $2^{3}$ .

$\ln (\frac{{(2^{3})}^{\frac{1}{3}}}{{(x^{3})}^{\frac{1}{3}}})$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (\frac{2^{3 (\frac{1}{3})}}{{(x^{3})}^{\frac{1}{3}}})$

Cancel the common factor of $3$ .

Cancel the common factor.

$\ln (\frac{2^{3 (\frac{1}{3})}}{{(x^{3})}^{\frac{1}{3}}})$

Rewrite the expression.

$\ln (\frac{2^{1}}{{(x^{3})}^{\frac{1}{3}}})$

Evaluate the exponent.

$\ln (\frac{2}{{(x^{3})}^{\frac{1}{3}}})$

Step 6

Simplify the denominator.

Multiply the exponents in ${(x^{3})}^{\frac{1}{3}}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (\frac{2}{x^{3 (\frac{1}{3})}})$

Cancel the common factor of $3$ .

Cancel the common factor.

$\ln (\frac{2}{x^{3 (\frac{1}{3})}})$

Rewrite the expression.

$\ln (\frac{2}{x^{1}})$

Simplify.

$\ln (\frac{2}{x})$

Write as a Single Logarithm 1/3( natural log of 64- natural log of 8x^3)

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